Question: $P(x)$ is a polynomial. Here are a few values of $P(x)$. $P(-6)=3$ $P(-4)=-1$ $P(4)=2$ $P(6)=-5$ What is the remainder when $P(x)$ is divided by $(x-6)$ ?
Solution: We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $P(x)$ is divided by $(x-{6})$ is $P({6})$, and we are given that $P({6})=-5$. In a similar manner, the remainder when $P(x)$ is divided by $(x+4)$, which can be rewritten as $(x-({-4}))$, is $P({-4})$, and we are given that $P({-4})=-1$. In conclusion, The remainder when $P(x)$ is divided by $(x-6)$ is $-5$. The remainder when $P(x)$ is divided by $(x+4)$ is $-1$.